Abstract
The aim of this work is to obtain quantum estimates for q-Hardy type integral inequalities on quantum calculus. For this, we establish new identities including quantum derivatives and quantum numbers. After that, we prove a generalized q-Minkowski integral inequality. Finally, with the help of the obtained equalities and the generalized q-Minkowski integral inequality, we obtain the results we want. The outcomes presented in this paper are q-extensions and q-generalizations of the comparable results in the literature on inequalities. Additionally, by taking the limit qrightarrow 1^{-}, our results give classical results on the Hardy inequality.
Highlights
We prove a generalized q-Minkowski integral inequality
The outcomes presented in this paper are q-extensions and q-generalizations of the comparable results in the literature on inequalities
[1 – p]q a x1–p Dq hq,r,a g (x) p–1 hq,r,ag(x) p–1–i hq,r,ag(qx) i i=0 dqx hq,r,ag(b)
Summary
We prove a generalized q-Minkowski integral inequality. With the help of the obtained equalities and the generalized q-Minkowski integral inequality, we obtain the results we want. The following definitions, notations and theorems for q-derivative and q-integral of a function f on [a, b] are given in [2, 3, 9]. We have the following properties of the q-integral of (2.4): x (I) Dq f (t) dqt = f (x).
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